1- Summary of Kinetic Theory of Gases
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1 Dr. Kasra Etmad Octobr 5, 011
2 1- Summary of Kntc Thory of Gass - Radaton 3- E4 4- Plasma Proprts
3 f(v f ( v m 4 ( kt 3/ v xp( mv kt V v v m v 1 rms V kt v m ( m 1/ v 8kT m 3kT v rms ( m 1/ E3: Prcntag of partcls havng spd btwn V 1 and V NV 1 V U 1 U rf ( U N
4 + A + A n n n a on ( m kt h 3 3/ on a xp( kt a n numbr dnsty of lctrons n on numbr dnsty of ons n a numbr dnsty of atoms m mass of lctron k Boltzmann constant T Tmpratur h Planck s constant Elctron charg on Partton functon of on a Partton functon of atom a E -E E onzaton = n 1/3 [V] E onzaton Ionzaton potntal of nutral atoms Lowrng of onzaton potntal E onzaton
5 P total = p molculs + p atoms + p ons + p tctons p = n k T For mult-tmpratur Plasmas: p = n kt + n on kt + n a kt a For qulbrum plasmas: p = (n a +n on +n a kt
6 Hydrogn n g a n a E xp( kt n E g a n a T k Numbr dnstyof atom n th nrgy stat Enrgy stat of th th nrgy lvl Dgnracy Partton functon Total numbr of atoms Tmpratur Boltzmann Constant
7 n g a n a E xp( kt n E g a n a T k Numbr dnstyof atom n th nrgy stat Enrgy stat of th th nrgy lvl Dgnracy n Partton functon onzaton E a g xp( 1 kt Total numbr of atoms Tmpratur Boltzmann Constant X atom Hydrogn E 4 =3E n=4 E 3 =E n=3 E =E n= E 1 =0 Ground Ground Stat Stat n=1 dgnracy g 4 =3 =1 g 3 = =1 g =1 E 1 =0 g 1 =1
8 Enrgy=3E X atom E 4 =3E E 3 =E E =E n=4 n=3 n= E 1 =0 g 1 =1 Ground Stat n=1 dgnracy g 4 =3 g 3 = g =1 3E1 E1 E1 0
9 Enrgy=3E X atom E 4 =3E E 3 =E E =E n=4 n=3 n= E 1 =0 g 1 =1 Ground Stat n=1 dgnracy g 4 =3 g 3 = g =1 3E1 E1 E1 0 3E1 E1 E1 0
10 Enrgy=3E 3E1 3E1 E1 E1 E1 E E1 E 4 =3E n=4 dgnracy g 4 =3 E1 E1 0 n g a n a a E xp( kt n onzaton 1 E g xp( kt E 3 =E E =E n=3 n= E 1 =0 g 1 =1 Ground Stat n=1 g 3 = g =1 3E1 E1 E1 0 3E1 E1 E1 0 3E1 E1 E1 0
11 Dr. Kasra Etmad Octobr 5, 0011
12 hc E(, T 5 Stfan-Boltzmann Law: 1 hc kt 1 E T Jm s 1 K 4 Wn Dsplacmnt Law: Max 310 T 7 ( s n Å
13 Contnuous Ln 4000 Å 7500 Å Contnuous Spctrum Absorpton Spctrum of Hydrogn Emsson Spctrum of Hydrogn Colors
14 lctron - + lctron - +, ( ( 1/ 8 T T N T T c ff T C fb T T g T, ( (1, (, (
15 Absorpton Emsson E=h E=h
16
17 Ej E=E j =h j E E E E total 1 n 4 1 n 4 E j a A g j j a E j 1 n 4 E xp( kt E contnum j j A j h A h j j j
18 Plasmas Optcally Thn Optcally Thck
19 Maxwllan Dstrbuton f ( v n nn n a g a m 4 ( kt Boltzmann Dstrbuton Saha Equaton n Planck s Functon a E xp( kt (mkt 3 h hc E(, T 5 3/ 3/ 1 hc kt v xp( xp( a 1 mv kt a kt T Maxwllan T Boltzmann T Saha T Planck
20 Complt Thrmodynamc Equlbrum T Maxwllan = T Boltzmann = T Saha = T Planck Local Thrmodynamc Equlbrum T Maxwllan = T Boltzmann = T Saha Partal Local Thrmodynamc Equlbrum T Maxwllan =T Saha
21 Exampl For an atmosphrc prssur hydrogn plasmas havng a tmpratur of 10,000 K dtrmn th followng: numbr dnsty of atoms, ons and lctrons Maxmum numbr dnsts of atoms, ons and lctrons that can contrbut to xctaton of hydrogn atoms from ground stat. Dnsty dstrbuton of nrgy stats of hydrogn atoms Intnsty [J/cm-3] of hydrogn spctral ln at Å (frst xctd to ground stat, g u =8, H =, A ul = s-1
22 EE 403/503 Introducton to Plasma Procssng Octobr 5, 011
23 Scond Law Mchancs Elctrcty and Magntsm Collsonal & radatv Procsss Equlbrum Macroscopc thory Atomc Structur Mcroscopc thory Equlbrum Thrmodynamcs, p, T, h, s, Flud dynamcs (Consrvaton Equatons Statstcal mchancs (Partton functon Kntc thory (Boltzmann Equaton Thrmodynamc Proprts.g. h=h(p,s Transport Proprts,, D,
24 Spcs dstrbuton functon f s (x,v,t Extrnal forcs Nt rat of ncras of spcs s, as a rsult of collsons btwn partcls of spcs s wth thos of spcs r t ( n s f s v. n s f s F m s s n s v f s C r sr Numbr of partcls n s Vlocty Mass of spcs s C sr r s ' ' fs( vs fr( vr fs( vs fr( vr v rsbdbdvr n n aftr th collson Impact paramtr
25 BoltzmannEquaton Drvaton? Drvaton? Drvaton? Maxwllan Dstrbuton Plasma proprts Consrvaton Equatons
26 Nglctng all xtrnal forcs n Boltzmann quaton gvs th Maxwll dstrbuton functon: dn 4n m f ( v v ( 3/ v xp( dv kt mv kt f(v Vlocts
27 Gnral Form Dpndnt varabl Spcfc to a partcular Manng of t ( ( u Unstady trm Convcton trm ( Dffuson trm S Sourc trm
28 Gnral: t ( ( u ( S = 1 t ( ( u 0 = u = m l = h t t t ( u ( uu ( m ( um l ( h ( uh ( u l ( m ( k c p p l S R h l S m
29 To utlz th consrvaton quatons n your applcaton, you nd: -a st of consrvaton quatons that ar rqurd to dscrbs your applcatons -St up your ntal condton and boundary condtons -Provd th rqurd thrmodynamc and transport proprts Solv th consrvaton quatons analytcally or numrcally
30 1- Dffusvty (Pag 49 Movmnt of gas partcls from hgh dnsty to low dnsty gas by random walk Partcl Flux Dffuson Coffcnt nv d Dn Fck s Law Dnsty Drft Vlocty 1 1 D vc v 3 3 Collson Frquncy For a monolthc gas: Avrag Vlocty 1 3 v Man Fr Path D n n x f 1 1 L mt Cross scton for bnary hard-sphr collsons
31 - Elctrcal Conductvty Currnt Dnsty J E [A/m ] Elctrcal Conductvty m n 3- Thrmal Conductvty Hat Flux q T Thrmal Conductvty f v Dgr of Frdom Hat Capacty For a monolthc gas: 5 64 fk kt m 1/ 3N Cvv Avag cross
32 4- Vscosty Forc rsultng from th nt transport of momntum from on rgon to anothr Coffcnt of Vscosty F 5- Moblty Drft Vlocty 1 3 U du dz d mv cross E [N/m ] [Ns/m ] Cross scton for th bnary hard-sphr collson Moblty Elctrc Fld 3 4 n m kt 1/
33 Lnk1 1- Intrnal Enrgy - Enthalpy 3- Entropy 4- Gbbs Fr Enrgy
34 g s s Es kt E s E transtonal E lctronc E vbratonal E rotatonal E onzaton transtonal lctronc vbratonal rotatonal onzaton
35 s kt E s s g 3/ 3 3 ( mkt h V dv h V kt mv al transton onzaton rotatonal l vbratona lctronc al transton kt E kt E lctroncs n j 1 ( For Hydrogn kt onzaton
36 For molcul mad of two dntcal atoms (rgd rotator: h 1 j( j 1 / kt 8 rotatonal ( j 1 j0 8 kt h Approxmat Soluton For harmonc oscllaton oscllaton 0 h ( 1/ kt 1 snh( h / kt
37 g s s Es kt Total partton functon for a systm composd of N dntcal, Indstngushabl partcls: total N N!
38 U kt N t (ln total Intrnal Enrgy Sum of nrgs of ndvdual atoms or molculs S k N ln total U T Entropy Dfnton: dq ds T S k ln W H U P V Enthalpy Dfnton: H=U+pV Probablty of systm N G H TS Gbbs Fr Enrgy Dfnton: G=U-TS+pV
39 In a Complt Thrmal Equlbrum (CTE systm th followng condtons ar mt. 1- Radaton mttd by th systm follow Black Body Radaton -All spcs hav Maxwllan dstrbuton 3- Kntc qulbrum must xst,., T =T h (E/P s small, T s hgh 4- Collson procss stablsh xctaton and onzaton qulbrum 5- Spatal varatons ar small Local Thrmodynamc Equlbrum (LTE mans that condtons -5 ar mt. Partal Local Thrmodynamc Equlbrum (PLTE mans that on condton s only partally valdatd (.g. xctaton may not xactly follow th Boltzmann dstrbuton.
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